From Relations to Geometry

Let $X$ be a finite set of $N$ entities and let $r:X\times X\to\{0,1\}$ be a binary relation. Instead of assigning a geometry to $X$ in advance, we ask whether the relation itself can generate one.

Two Views of a Relation

Fixing the first argument gives the family of propositions $p_x(y)=r(x,y)$, denoted by $\mathcal P_r$. Fixing the second gives $q_y(x)=r(x,y)$, denoted by $\mathcal Q_r$. The first family observes incoming relational behavior, while the second observes outgoing relational behavior.

The two families generate two topologies, $\tau_r^H$ and $\tau_r^T$. Using both sides gives the join topology $\tau_r=\tau_r^H\vee\tau_r^T$. For example, if $r(x,y)$ means that animal $x$ eats animal $y$, the head topology compares animals according to which animals eat them, while the tail topology compares them according to which animals they eat.

From Topology to Distance

Each entity has an incoming binary profile and an outgoing binary profile. Concatenating them gives $\Phi_r(x)\in\{0,1\}^{2N}$. The natural Hamming distance is $d_r(x,y)=d_r^H(x,y)+d_r^T(x,y)$. Two entities have distance zero exactly when neither $\mathcal P_r$ nor $\mathcal Q_r$ can distinguish them. Thus $d_r$ is a pseudometric on $X$ and becomes a metric after identifying entities with identical joint profiles.

A System of Relations

For $m$ relations, concatenate the head and tail profiles produced by every relation, giving $\Phi(x)\in\{0,1\}^{2mN}$. The topology of the whole system is the join of all head and tail topologies, while the natural distance is Hamming distance on the joint profile.

Compressing the Geometry

For binary vectors, $d_H(u,v)=\|u-v\|_2^2$. Thus the relational Hamming geometry has an exact squared-Euclidean realization in dimension $2mN$. The Johnson–Lindenstrauss lemma allows the $N$ entity profiles to be projected into $\mathbb R^{O(\varepsilon^{-2}\log N)}$ while approximately preserving all pairwise Euclidean distances, and hence the corresponding Hamming geometry through the squared-distance relation.

The complete picture is $\text{Relations}\longrightarrow\text{Propositions}\longrightarrow\text{Topology}\longrightarrow\text{Hamming Geometry}\longrightarrow\text{Low-Dimensional Geometry}$. The embedding is therefore not the starting point: it is a compressed representation of a geometry already generated by the relational system.

Please carefully check the arguments on your own if you plan to use them in your research.